1. Statement of the Technical Field
The invention concerns cryptographic systems. More particularly, the invention concerns cryptographic systems implementing a method for digitally generating a chaotic numerical sequence.
2. Description of the Related Art
Chaotic systems can generally be thought of as systems which vary unpredictably due to the defining characteristics of: sensitivity to initial conditions; being mathematically dense; and being topologically transitive. The characteristics of denseness and topological transitivity loosely mean that the resultant numerical values generated by a chaotic circuit do not clump together, yet take every feasible value in the range. Chaotic systems are also distinguished by a sensitive dependence on a set of initial conditions and by having an evolution through time and space that appears to be quite random. When measured or observed, chaotic systems do not reveal any discernible regularity or order. However, despite its “random” appearance, chaos is a strictly deterministic evolution.
There are many types of chaotic cryptographic systems known in the art. Such chaotic cryptographic systems include a chaotic based encryption system and a chaotic based decryption system. Chaotic cryptographic systems offer promise for being the basis of a next generation of secure waveforms, providing low probability of Exploitation (LPE). Chaotic systems are typically comprised of analog circuits implementing chaos generators. Cryptographic systems are typically based on pseudo-random number generators driving mappings in finite algebraic structures.
Chaos generators have been conventionally constructed using analog chaotic circuits. The reason for reliance on analog circuits for this task has been the widely held conventional belief that efficient digital generation of chaos is impossible due to the inherent sensitivity to initial conditions dictating impractical wordwidths. Notwithstanding the apparent necessity of using analog type chaos generators, that approach has not been without problems. For example, analog chaos generator circuits are known to drift over time. The term “drift” as used herein refers to a slow variation in one or more parameters of a chaotic signal.
Prior art cryptographic systems may use multiple pseudo-random number generators to generate exceedingly complex pseudo-random sequences. However, such cryptographic systems only produce more complex pseudo-random number sequences that still possess even more complex pseudo-random statistical artifacts and no true chaotic properties. The sequences become more difficult to unravel and near impossible to exploit as the mappings become more complex. While certain polynomials can mimic chaotic behavior, the arithmetic precision required to generate chaotic number sequences required an impractical implementation. Stated differently, the binary arithmetic necessary in order to achieve digital chaos was prohibitive.
In view of the forgoing, there is a need for a chaotic cryptographic system configured to generate a sequence having chaotic properties. There is also a need for a method for digitally generating a chaotic number sequence that can be used in a variety of cryptographic system applications.